Spectral cut-off regularizations for ill-posed linear models

E Chernousova (1) , Yu Golubev (2)

1 MIPT - Moscow Institute of Physics and Technology [Moscow]
2 I2M - Institut de Mathématiques de Marseille

This paper deals with recovering an unknown vector β from the noisy data Y = Xβ + σξ, where X is a known n × p-matrix with n ≥ p and ξ is a standard white Gaussian noise. In order to estimate β, a spectral cutoff estimate β(m,Y) with a data-driven cutoff frequency m(Y) is used. The cutoff frequency is selected as a minimizer of the unbiased risk estimate of the mean square prediction error, i.e. m(Y) = arg min_{m}\| Y − X β (m, Y)\|^2 + 2σ^2 m. Assuming that β belongs to an ellipsoid W, we derive upper bounds for the maximal risk sup_{β∈W}E \|β[m(Y), Y] − β\|^2 and show that β[m(Y), Y] is a rate optimal minimax estimate over W.

Mots clés: iIll-posed linear model; spectral cut-off regularization; data-driven cut-off frequency; oracle inequality; minimax risk; sec-ondary 62C10

Domaines: Statistiques [stat]

E Chernousova, Yu Golubev. Spectral cut-off regularizations for ill-posed linear models. Mathematical Methods of Statistics, 2014, ⟨10.3103/S1066530714020033⟩. ⟨hal-01292417⟩